Using the cube root table, find the cube root of 85.9 and 309400.
(a)4.412 and 67.643
(b)4.612 and 67.643
(c)4.412 and 67.543
(d)4.412 and 67.613

Hint:
To find the cube roots of the numbers which are not in the cube root table, we will find the numbers which are just greater than and just smaller than the given number and are in the cube root table. After that we will use the unitary method to evaluate the cube root of the required number.

Complete step-by-step answer:
We need to use the cube root table to find the cube roots of 85.9 and 309400.
First, we will find the cube root of 85.9
The cube root table gives us the cube roots for integers up to 9900.
So it does not give us the cube root of 85.9
So, we will use cube roots of 85 and 86 to find cube root of 85.9
From the cube root table, we have:
$\sqrt[3]{85}=4.397$ and $\sqrt[3]{86}=4.414$
Since, $85<85.9<86$
Hence, $\sqrt[3]{85}<\sqrt[3]{85.9}<\sqrt[3]{86}$
$4.397<\sqrt[3]{85.9}<4.414$\[\]
Now we see that, for the difference of ( 86 - 85 ) i.e. 1, we have, The difference in the cube root values = $4.414-4.397=0.017$
For the difference of ( 85.9 - 85 ) i.e. 0.9, we have the difference in cube root values:
$\dfrac{0.017}{1}\times 0.9=0.015$ (up to three decimal places)$\sqrt[3]{85.9}=4.412$(using unitary method)
 Hence, $\sqrt[3]{85.9}=\sqrt[3]{85}+0.015$
$\sqrt[3]{85.9}=4.397+0.015=4.412$
Hence,$\sqrt[3]{85.9}=4.412$
Now, we will find the cube root of 309400
The cube root table gives cube roots of natural numbers up to 9900. Clearly, 309400 is greater than 9900. So, we write \[309400=\text{ }1547\text{ }\times \text{ }200\]
Taking cube root, we get the following:
\[\sqrt[3]{309400}=\text{ }\sqrt[3]{1547}\text{ }\times \text{ }\sqrt[3]{200}\] …(1)
We will first calculate $\sqrt[3]{1547}$
Now, 1500 < 1547 < 1600
$\sqrt[3]{1500}<\sqrt[3]{1547}<\sqrt[3]{1600}$\[\]
From the cube root table, we have $\sqrt[3]{1500}<\sqrt[3]{1547}<\sqrt[3]{1600}$
$\sqrt[3]{1500}=11.45$ and $\sqrt[3]{1600}=11.70$
Thus, for the difference of ( 1600 - 1500 ) i.e. 100, we have,
The difference in the cube root values = 11.70 - 11.45 = 0.25
For the difference of ( 1547 - 1500 ) i.e. 47, we have the difference in the cube root values =
$\dfrac{0.25}{100}\times 45=0.117$ (up to three decimal places) (using unitary method)
 Hence, $\sqrt[3]{1547}=\sqrt[3]{1500}+0.117$
$\sqrt[3]{1547}=11.45+0.117=11.567$
Also, from the cube root table, we have:
\[\sqrt[3]{200}=5.848\]
Now, substituting these values in (1), we get:
\[\sqrt[3]{309400}=\text{ }\sqrt[3]{1547}\text{ }\times \text{ }\sqrt[3]{200}\]
\[\sqrt[3]{309400}=\text{ }11.567\times 5.848=67.643\]
 Hence, \[\sqrt[3]{85.9}=4.412\]and \[\sqrt[3]{309400}=67.643\]
So, option (a) is correct.

Note: We can use the given options to our benefit and solve the question easily and quickly. Initially, we found that \[4.397<\sqrt[3]{85.9}<4.414\]. So option (c) will be eliminated. The rest of the options have the same value and so we can conclude that \[\sqrt[3]{85.9}=4.412\]. Similarly, we can use the options to find \[\sqrt[3]{309400}\] comparatively faster.